For any given mixed state and observable, it has been shown that the
minimal averaged variance among all possible pure-state
decompositions is exactly the quantum Fisher information and the
maximal averaged variance is the variance itself
[1]}, [2]}, [3]}, [4]}. Concerning the
relation between the variance and quantum Fisher information, the
improved uncertainty relations have been derived in terms of quantum
Fisher information and variance [5]}, [6]}, [7]}.
Moreover, the variance-based uncertainty relation has been
incorporated into quantum multiparameter estimation by investigating
the quantum Fisher information matrix and the classical Fisher
information matrix from measurements [8]}.
It is well known
that the quantum Fisher information places the fundamental limit to
the accuracy for estimating an unknown parameter and plays an important role in quantum metrology.
The uncertainty
relation and quantum metrology are then closely connected by the
variance and quantum Fisher information.
